Bell violation for unknown continuous-variable states

We describe a new Bell test for two-particle entangled systems that engages an unbounded continuous variable. The continuous variable state is allowed to be arbitrary and inaccessible to direct measurements. A systematic method is introduced to perform the required measurements indirectly. Our results provide new perspectives on both the study of local realistic theory for continuousvariable systems and on the non-local control theory of quantum information. S Online supplementary data available from stacks.iop.org/NJP/16/013033/ mmedia The issue of incompatibility between local realism and the completeness of quantum mechanics was originally raised for unbounded continuous variables in two-party systems by Einstein et al [1]. Experiments to test local realism based on inequalities proposed by Bell [2] and his followers [3] imply, as is well known, that classical realism must be discarded as the basis for a universal theory. This has been repeatedly demonstrated in experiments with discrete variable systems [4–8]. Methods for testing local realism in continuous-variable systems have been proposed in order to advance the goal of reaching a completely loophole-free conclusion, and experimental tests on continuous-variable systems have been carried out [9–13]. However, these tests and all continuous-variable proposals to date [9–20] fall short because they rely on advance knowledge 1 Author to whom any correspondence should be addressed. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New Journal of Physics 16 (2014) 013033 1367-2630/14/013033+08$33.00 © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft New J. Phys. 16 (2014) 013033 X-F Qian et al of the state under test. These methods fail whenever the state under test is unknown because then there is no basis by which measurement strategies can be guaranteed effective. One reason is that non-local correlations present in the original state can evade detection under dimensional reduction [21], as may happen, for example, in pursuing pseudo-spin [13, 18, 19] or binning [20] methods. An exceptional approach by Cavalcanti et al [22] leads to a continuous multipartite inequality that does not rely on advance knowledge of the state under test. However, to construct their inequality, operator commutation relations must be ignored, which also eliminates a large category of local realistic theories from test—Sun et al [22]. Additionally, violation of these inequalities may not be possible with only two parties—Salles et al [22]. Thus two obstacles that have not yet been overcome are these: to derive a standard Bell–Clauser, Horne, Shimony and Holt (CHSH) inequality [3] for an arbitrary and unknown bipartite input state in an unbounded continuous-variable state space, and to describe a currently feasible experimental method for its test. There are significant fundamental and practical reasons for solving this problem. On the fundamental side, a clear understanding of the domains of continuous-variable space which are incompatible with local realism remains to be achieved. More practically, in recent years paradigm-shifting quantum technologies have been developed which depend upon Bell non-locality in theory, and in some cases require the experimental violation of a Bell inequality of an unknown state [23]2. Methods which permit Bell–CHSH inequalities to be formed and then tested on unknown states in continuous-variable systems may aid in the development and implementation of these technologies. In this paper we take a significant step towards overcoming both obstacles. To provide easy visualization, we address both issues in a specific scenario using the following two-photon down-conversion state: |ψAB〉 = cos θ |H〉A ⊗ ∫ d E q h( E q)| E q〉B + sin θ |V 〉A ⊗ ∫ d E q v( E q)| E q〉B, (1) where |E q〉B is one of a continuum of delta-normalized one-photon transverse momentum states of photon B, and |H〉A and |V 〉A denote horizontally and vertically polarized quantum states of photon A. We assume that the transverse momentum state of photon A and the polarization of photon B factor out of the quantum state and therefore need not be indicated. The sin θ and cos θ factors are included in writing |ψAB〉 to preserve its unit normalization, as the complex continuum amplitudes h( E q) and v( E q) are assumed to be unit-normalized, i.e. ∫ d E q |h( E q)|2 = ∫ d E q |v( E q)|2 = 1. Beyond normalization, nothing else is assumed about h( E q) and v( E q), including the value of their generally non-zero scalar product ∫ d E q h( E q)v( E q)≡ z 6= 0. (2) The two-photon state in (1) has an important freedom in the amplitude functions h( E q) and v( E q), which are arbitrary superpositions of the modes in continuous E q space. In the following we will use the term bundle to refer to an arbitrary superposition of | E q〉 states. Note that this means that it is impossible to fully determine the state (infinitely many measurements would be required). This point is crucial because it is the stopping point for attempts up to the present time to fully engage a continuous degree of freedom in Bell inequality analysis. We have overcome this roadblock, as we describe below. 2 These technologies include quantum-assisted communication complexity, quantum-assisted zero-error communication, device-independent quantum key distribution and device-independent randomness generation. For a brief introduction to these topics see Brunner et al [23].